Question

The kinetic energy K of a rotating body depends on its moment of inertia I and its angular speed $\omega$. Assuming the relation to be $K=k{I}^a{\omega }^b$ where k is a dimensionless constant, find a and b. Moment of inertia of a sphere about its diameter is $\frac{2}{5}M{r}^2$.

Answer 1

$K=K{I}^a{\omega }^b$, where K = kinetic energy of totating body and k = dimensionless constant

Dimensions of left side are $K=\left[M{L}^2{T}^{-2}\right]$

Dimensions of right side are $I^a=[M{L}^2{]}^a,{\omega }^b=[T^{-1}{]}^b$

Equating the dimensions of both sides, we get

2 = 2a and - 2 = - b

a = 1 and b = 2